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GUIDED WAVE BASED NONDESTRUCTIVE TESTING AND
EVALUATION OF EFFECTIVE ELASTIC MODULI OF LAYERED
COMPOSITE MATERIALS
E.V. Glushkov*, N.V. Glushkova, A.A. Eremin
Institute for Mathematics, Mechanics and Informatics, Kuban State University,
Stavropolskaya St., 149, Krasnodar 350040, Russia
Abstract. The paper presents a nondestructive technique for the evaluation of elastic moduli
of layered fiber-reinforced composites. It is based on the minimization of the discrepancy
between the theoretically calculated and experimentally measured dispersion curves of guided
Lamb waves generated by surface-mounted piezoelectric wafer active sensors. Numerical
simulation is performed in the context of 3D linear elastodynamics for anisotropic solids. It
relies on the explicit integral and asymptotic representations derived for the force-generated
wave fields in terms of Green’s matrix of the composite structure considered. The
minimization of the least-square error sum between the measured and calculated data is
achieved through a real coded micro-genetic algorithm. The reliability of the proposed
approach is experimentally validated via static tensile tests as well as by the frequency
response evaluation.
1. Introduction
Nowadays thin-walled engineering structures made from layered composite materials become
widespread in aerospace applications, automotive and shipbuilding industry, civil
engineering, etc. While in service, the composites are subject to various types of negative
environment impacts, mechanical shocks, damage accumulation, fatigue and aging, which
lead to the degradation of their useful mechanical properties. Consequently, it is of great
importance to develop reliable non-destructive evaluation (NDE) techniques capable of
damage detection and identification as well as of a continuous monitoring of composite’s
elastic properties during the operation in a fast and non-intrusive manner [1].
One of the modern approaches, which addresses these problems, is the utilization of
elastic guided waves (GW), namely, Lamb or SH-waves [2]. The waves of this type can
propagate over long distances from the excitation source, interacting with any
inhomogeneities and allowing for their localization and assessment through the analysis of
scattered wave fields. Successful GW application in NDE systems requires reliable
mathematical models and simulation tools for the evaluation of GW amplitude and dispersion
characteristics. For multilayered anisotropic composites, the latter depend not only on the
frequency but also on the propagation direction, resulting in complex wave patterns and
complicating damage detection.
Guided waves can be also used for the evaluation of effective elastic moduli of
composite materials, which are an indispensable input for computer models of ultrasonic
NDE [3, 4]. The solution of such an inverse problem can be used for a noninvasive
Materials Physics and Mechanics 23 (2015) 56-60 Received: March 27, 2015
© 2015, Institute of Problems of Mechanical Engineering
assessment of the degradation of mechanical properties during the lifetime monitoring of
composite structures. Moreover, such algorithms may be employed for the investigations of
the temperature and prestressed state influence on the material elastic moduli [5].
The reconstruction method, proposed below, is based on the minimization of
discrepancies between the theoretically calculated and experimentally measured GW
dispersion characteristics (group velocities) of the fundamental antisymmetric Lamb wave –
A0 mode. The developed algorithm is described in the second section while the results of its
experimental implementation and verification are presented and discussed in the third part of
the paper.
2. Algorithm for determining elastic moduli
Layered fiber-reinforced composite materials with unidirectional [0o] or cross-ply [0
o, 90
o]s
lamination schemes of elementary unidirectional prepregs are considered. Within the
computer models below, the specimens are assumed to consist of transversely isotropic
sublayers with identical elastic properties and known density and thicknesses. The task is to
determine five effective elastic moduli of the prepreg: C11, C12, C22, C44, and C55. It is also
assumed that the fibers of the top layer are aligned along the Ox axis, thus, the principal axes
of the both types of composites coincide with the horizontal Cartesian coordinate
axes (Fig. 1).
Fig. 1. Problem geometry and experimental setup sketch.
In a low frequency range, the phase and group velocities of A0 mode propagation in the
directions of principal axes depend only on the specimen’s thickness, density and
corresponding Young moduli (elastic constants C11 and C22) [6]. At the same time, for mid
and high frequencies, the A0 dispersion characteristics are mainly influenced by shear elastic
moduli (constants C44 and C55). To illustrate both these statements, Fig. 2 depicts the relative
deflection ,0 ,0100% ( ) /g g gv v v of the group velocity calculated as a function of
frequency in the course of each aforementioned elastic modulus variation. Here vg,0 is the A0
group velocity at frequency f [kHz] calculated for the pristine test unidirectional specimen of
thickness h=1 mm with the prepreg elastic properties from Ref. [7]: C11= 130.76, C12= 5.26,
C22= 12.99, C44= 3.75 and C55= 5.97 [GPa]; ρ=1578 kg/m3. The velocities vg are acquired
through the ±30% variation of the elastic modulus specified in the corresponding subplot.
The proposed algorithm consists of the following steps:
1. Guided waves are generated by the adhesively bonded piezoactuator excited with the
windowed tone-bursts of varying central frequencies. Further, laser vibrometer is used for
measuring out-of-plane velocities of propagating waves at the points located on the plate
surface along the direction of upper sublayer fiber alignment (points Ai) and in the
perpendicular one (points Bi), i=1,2,3 (Fig. 1).
2. Measured signals are processed with the continuous wavelet transform with the Gabor
mother wavelet [8]. Then, the time of flight (ToF) of the wave package at each local
57Guided wave based nondestructive testing and evaluation of effective elastic moduli ...
frequency is extracted by using the corresponding magnitude peaks of the wavelet
coefficients. The A0 group velocity is approximated as the ratio of the distances between Ai (or
Bi) points to the time of traveling wave propagation between these points. The values obtained
along every particular direction are then averaged.
Fig. 2. Frequency sensitivity of A0 mode group velocity to material constants variation for the
unidirectional plate.
3. As soon as the experimental group velocity dispersion curves are obtained for the both
directions of A0 propagation considered, the elastic modulus reconstruction procedure is
reduced to a minimization problem for the ravine objective function
2
, ,
1
(C) 1N
c m
g j g j
j
E v v
, (1)
where С is a candidate matrix of material constants (a varied input of the optimization
problem); ,
c
g jv and ,
m
g jv are computed and measured group velocities for the propagation
directions considered, N is the total amount of measured velocities. To obtain ,
c
g jv , the
algorithm of Green’s matrix computation for a multilayered arbitrarily anisotropic structure
[9] and asymptotic representations for group velocities of quasi-cylindrical guided waves
excited in a laminate anisotropic waveguide by a localized surface load [10] are used.
4. Objective function (1) is minimized using a real coded microgenetic algorithm with a
simulated binary crossover [11]. Every individual chromosome represents a candidate
solution (matrix C), while its genes are specific elastic moduli.
Preliminary numerical testing of this approach on unidirectional and cross-ply laminates
with known prepreg elastic properties has shown its high numerical efficiency and
considerable noise stability. For example, with the input array of N=32 measured velocities
(16 values of A0 mode group velocities for each principal axis) subjected to 5 % artificial
Gauss-type noise, the maximum relative discrepancy of the reconstructed elastic moduli is
less than 5 % for unidirectional laminate and does not exceed 11 % for the cross-ply one.
3. Experimental verification of the algorithm
The developed approach has been experimentally tested on carbon fiber-reinforced plastic
plates using the equipment facilities of Institute for Mechanics, Helmut Schmidt University,
Hamburg, Germany. The measurements have been performed for composite samples with
unidirectional [0o]4 and cross-ply [0
o, 90
o]s lay-ups. The specimens’ dimensions are
1000×1000×1.13 mm3, and the prepreg’s density is 1482 kg/m
3; fiber volume fraction is
about 52 %. Guided waves are generated by piezoelectric wafer active sensors adhesively
58 E.V. Glushkov, N.V. Glushkova, A.A. Eremin
bonded to the composite plate, while the out-of-plane surface velocities are acquired with the
1D scanning laser Doppler vibrometer Polytec PSV-400.
The results obtained for both unidirectional and cross-ply laminates after the averaging
over eight genetic algorithm runs are summarized in Table 1.
Table 1. Effective elastic properties of the transversely isotropic prepreg evaluated with the
proposed approach (all the values are given in GPa).
Plate type C11 C12 C22 C44 C55 Ex Ey
Unidirectional laminate 110.5 4.23 11.8 2.59 3.71 108.6 8.05
Cross-ply laminate 106.6 3.15 8.11 2.46 3.91 104.8 6.81
To verify the obtained results, standard tensile tests have been performed on the
coupons cut from the unidirectional sample. The following values of Young moduli along the
principal axes have been obtained: Ex =112.4 GPa and Ey =7.69 GPa. These values are in
good agreement with the restored effective elastic moduli specified in the last two columns of
Table 1. A discrepancy in Ey values obtained for unidirectional and cross-ply plates are
explained by a relatively low sensitivity of A0 mode group velocity to this constant variation
for a cross-ply laminate compared to the other moduli.
Fig. 3. Measured (dashed lines) and theoretically restored (solid lines) dispersion curves for
A0 and S0 mode wavelengths along the fibers (top subplots) and in the perpendicular direction
(bottom subplots).
As an additional test, frequency dependencies of fundamental symmetric and
antisymmetric mode wavelengths calculated using the elastic constants from the first row of
Table 1 have been compared with corresponding experimental values. The latter have been
obtained by applying time and spatial Fourier transforms to the out-of-plane velocities
scanned along the composite’s principal axes. Theoretical and experimental results are in
good coincidence for both types of laminates (Fig. 3).
Moreover, the A0 wavelengths measured at low excitation frequencies for the
unidirectional composite have been utilized for an approximate estimation of Young’s moduli
Ex and Ey via the relation from Ref. [6] 4 2 2 2
, 3 ( ),x y pE v H f where pv f , λ [m] is the
A0 wavelength for the corresponding direction, f [Hz] is the oscillation frequency, and H [m]
is the waveguide thickness. The obtained results (Ex=97.8 ГПа and Ey=7.54 ГПа) are in good
agreement with the values presented in Table 1.
4. Conclusions
A method of non-destructive evaluation of effective elastic moduli of fiber-reinforced
laminate composite plates based on experimentally obtained group velocity dispersion curves
59Guided wave based nondestructive testing and evaluation of effective elastic moduli ...
of the fundamental antisymmetric Lamb mode has been developed. Its efficiency is confirmed
by the conventional tensile test results as well as by the comparison of the predicted
dispersion characteristics, calculated with the restored elastic moduli, with the experimental
data.
Acknowledgements
This work is partly funded by the RFBR (projects No. 13-01-96516 and 14-08-00370) and the
Ministry of Education and Science of the Russian Federation (agreements No. 1.189.2014K
and 11.9165.2014). The authors highly acknowledge the support of Prof. R. Lammering and
his team members (T. Fedorkova and S. Föll) (Helmut Schmidt University, Hamburg) in laser
vibrometry experiments.
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60 E.V. Glushkov, N.V. Glushkova, A.A. Eremin